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Rigid Rogowski coils & a calibrator

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klugesmith

Wed Jan 20 2010, 03:56AM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1714

Introduction -- 2010 Jan 19
I first played with Rogowski coils four years ago, while dabbling in can-crushing after 10 kJ capacitor arrived. Got back into the subject a couple months ago, because of an unprecedented opportunity at work.

Here's my original coil, wound on a thick nylon washer. Two interleaved toroidal windings, one right- and one left-handed, go all the way around the core. They are electrically connected in series. So the net number of turns -around- the primary conductor is zero, just as in a flexible Rogowski coil wound on a coaxial cable dielectric. (That feature helps to minimize the sensitivity to primary currents that are not "encircled" by the sense coil.)

I calculated and measured the sensitivity of my coil, before ever visiting this forum. Result of that preliminary calibration measurement: M = 37 nH.

About Rogowski coils:
ANY roughly toroidal sense coil will develop a voltage proportional to di/dt, without saturation at high currents, if it has no ferromagnetic core and lies close to a primary conductor.
Ideally, the mutual inductance (sensitivity) is:

flat (constant) for any primary current path that passes "inside" the toroid,

zero for any primary current path that passes "outside".

To meet both criteria within a few percent is an engineering challenge, and is critical in applications such as metering for revenue. In flexible coils wound on co-ax, the place where the dead end meets the live one (to borrow a term from rope-work) is a trouble spot. Where, exactly, is the boundary between "inside" and "outside"?

Some vendors make both flexible and rigid Rogowski coils. The latter are represented as more accurate, but lack the convenience of clipping around a fixed wire or bus bar. A relatively recent development is rigid coils made as PC boards, for geometric symmetry and economical fabrication. This example,

, stacks two PC boards "wound" in opposite directions.

My printed circuit coil
Fast forward to late 2009. I became involved in the design of a round PC board that is 1/4 inch thick, with 30 layers. It's more than 20 inches in diameter, with more than 33,000 plated-through holes -- a new record for me & my co-workers. And it has a round hole almost 8 inches in diameter! That means each one produces a scrap of thick, multilayer PC board. ( Not unlike Corian and granite scraps when fancy kitchen counter-tops are prepared for sinks).

Well, I got approval to put a little Rogowski coil in the cut-out piece. Without time to run the numbers, I made it geometrically similar to the hand-wound coil. The right-hand winding uses top layer & one close to the bottom; left-hand winding uses bottom layer & one close to the top.

Soon I will get the fabricated board, do the calculations and measurements, and report the results.
Then polish & report on an inexpensive calibrator to generate a substantial, well-known current at the frequency of interest.

Sensitivity Formulas (2010 Jan. 21)

Steve McConner wrote ...
...There's a simple equation for Rogowski coil output, if I remember right it's just
Vout = u0 x N x A x di/dt (A is the loop area of one turn of the Rogowski coil)

Correct, where N is the linear winding density in turns per meter (not the total number of turns, which would be dimensionally wrong).
Combining the Rogowski coil sensitivity factors gives: Vout = M x di/dt,
where M = mutual inductance in henries = total flux linkage (webers) per ampere = u0 x N x A.
u0 = 4e-7 * pi H/m.

Consider this example from a 4hv-er:

(Are you here, aonomus? or TDU?)
I figure a coil of AWG30 wire closely wound (N = 3500 per meter)
on RG-6 cable dielectric (effective turn diameter 4.9 mm)
will have M of about 83 nH, regardless of how long a section is wound.
TDU's reported sensitivity of coil at

works out to M = 10.4 nH.

Here's a simple derivation.
At a distance r from the primary conductor, flux density B = u0 * H = u0 * i / ( 2 * pi * r ). (webers/m^2)
Each small turn of area A will have flux linkage F1 = A * B. (webers)
Total number of turns (in entire toroid) is Nt = 2 * pi * r * N.
Total flux linkage is Nt * F1 = u0 * N * A * i = M * i.

There's a different formula for coils whose turn size is not small compared to r. Suppose the toroid has inner and outer radii r1 and r2, and length L. Integrating the flux density over one rectangular turn, we get linkage F1 = i * u0 * L * ln(r2/r1) / (2*pi). See picture near bottom of

, or a better ref beginning at page 8 of:

Physical progress (2010 Jan 31)
Anyway, an update is in order. My Rogowski board came in, but I had to saw it out from round scrap -- no fancy outline or internal routing for the free G-job. It has a 3mm hole in the center, which will permit testing. Enlarging that to target ID of 1 inch will be a challenge -- could wear out a cheap hole saw on the fiberglass, but I think they tend to cut oversize & there isn't much room to spare.

The DC resistance is 0.8 ohms, and the voltage drop is divided among all turns as expected, confirming that there are no shorts or opens.

Meanwhile, I calculated the sensitivity. 39.6 nH for hand-wound coil and 32.1 nH for the PCB coil, by the logarithmic formula, and only a few percent less by the linear formula!

Here's a longitudinal section drawing, roughly to scale. The dash-dotted line is a conventionally represented centerline / axis of symmetry; showing the other half would waste pixels.

The logarithmic formula accounts for magnetic field variation within the area of a turn, while the linear formula takes it to be constant at an average radius (the radius at which we compute N in turns per meter). Here's how the linear formula diverges as a coil with rectangular turns becomes "thick":

The r2/r1 ratio has to reach 3 (width/average_radius = 1) before error reaches 10%. For my handwound coil, r2/r1 is 1.88 (w/r = 0.61) and the discrepancy is only 3%. For the PCB coil it's less than 2%. For a similar-diameter coil wound on RG6 coax, with w/r < 0.25, the linear formula gets within 0.5 %. Errors in estimating the effective dimensions are probably greater than that, in all these cases.

Steve McConner wrote ...
The OLTC primary could also be used as a calibrator, because the peak primary current could be calculated from the capacitance, charging voltage, and the measured resonant frequency.

Yup, that's the idea -- use the linear system under test, and calculate its di/dt. That's what I did before with can-crusher setup: discharge from about 10 volts, and determine the damped period and damping ratio by measuring a couple of details in scope waveform. Knowing C, can find L, R, and the current waveform. It gets interesting when damping is significant: the phase shift beteeen v, i, and di/dt is less than 90 degrees. What I am prepared to add this month is 1) compactness and 2) automatic repetition to facilitate use with an analog oscilloscope.

Progress with calibrator (2010 May 12)
Early results from the new calibrator agree with last year's measurements on the actual can crusher. Roughly, C = 50 uF, L = 2 uH, Z0 = 0.2 ohms, R = 0.02 ohms, f = 16 kHz, and discharge from 9 volts gives peak current of about 40 amps.

The real system is inconvenient for RLC parameter measurements, not to mention Rogowski coil calibration, because:
1. Capacitor weighs 150 lbs -- hard to carry it to good instruments at work.
2. Mechanically triggered spark gap, at low voltage with zero gap, exhibits contact bounce on many shots. That causes exceptionally ugly di/dt waveforms in the first few microseconds. I got acceptable shots every few tries.

Remedy for 1: Build a good lower-voltage C of the same value, using a bundle of plastic film capacitors. (avoid ceramic because of temperature and voltage coefficients). Can still use real can-crushing or coin-shrinking coils.
Remedy for 2: mercury-wetted relay (a traditional low-on-resistance fast bounceless switch

or a MOSFET (which, unlike BJT / SCR / IGBT, is low resistance near zero volts in both directions).

Pictures, schematics, scope pictures, and analysis are on the way.

Steve Conner

Wed Jan 20 2010, 10:12AM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

Klugesmith wrote ...

It's more than 20 inches in diameter, with more than 33,000 plated-through holes -- a new record for me & my co-workers. And it has a round hole almost 8 inches in diameter!

Good grief, that's hardcore! A test fixture for some advanced IC?

Anyway, I used Rogowski coils to sense primary currents in my OLTCs, and got good results with them. I just wound magnet wire onto a piece of flexible plastic pipe, coveed it with heatshrink, passed the dead end back through the pipe to avoid the "single turn effect", and then stuck the ends together with a pipe fitting. I used a passive integrator built into the coil with a RC time constant 10x longer than the lowest frequency I wanted to measure.

The OLTC primary could also be used as a calibrator, because the peak primary current could be calculated from the capacitance, charging voltage, and the measured resonant frequency. My mini coil did about 750A peak, and my big one 4kA.

There's a simple equation for Rogowski coil output, if I remember right it's just

Vout = u0 x N x A x di/dt (A is the loop area of one turn of the Rogowski coil)

and with an integrator:

Vout = (1/RC) (u0 x A x I)

Dr. Slack

Wed Jan 20 2010, 10:24AM

Registered Member #72 Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659

Nice pre-emptive use of your employer's process!

When they go to version 2 of the board, add another version of the Rogowski in the middle. This is for a "Rogowski clamp". I've never seen one described, so I ought to be quiet about it until I've done a patent search, but WTH.

Consider two half donuts, each on twisted pair flexible. In series, postioned carefully together, they are electrically and mechanically identical to a full donut, so should work exactly the same way. However, you can split them and place them round a cable. Obviously the rejection of external fields will be degraded slightly by the departure from uniform windings at the ends of the two half coils. It should be possible to compensate for a finite gap betwen the clamping faces by putting an extra turn on the ends, so making the average turns density more uniform when seen from a small distance

klugesmith

Thu Jan 21 2010, 07:35PM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1714

[edit 5/12/2010] consolidated to top post

klugesmith

Mon Feb 01 2010, 06:25AM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1714

[edit] consolidated to top post

Patrick

Thu Apr 15 2010, 03:06AM

Registered Member #2431 Joined: Tue Oct 13 2009, 09:47PM
Location: Chico, CA. USA
Posts: 5639

WHOA !!! i hope you put procedure and results, measurement and a how to guide together for the rest of us! this is so potentially useful for many of us. If i can figure out what the hell those formulas and math mean!

i would like to do this and the nylon washer ideawas greAT.

klugesmith

Thu May 13 2010, 07:40AM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1714

Patrick wrote ...

WHOA !!! i hope you put procedure and results, measurement and a how to guide together for the rest of us! this is so potentially useful for many of us. If i can figure out what the hell those formulas and math mean!
i would like to do this and the nylon washer ideawas greAT.

Thanks.
I just did some post consolidation (see OP).

My calibrator, like Steve's, uses an existing RLC tank circuit -- in this case periodically rung by closing a switch. The math is especially simple if damping is very low: R << sqrt(L/C). Starting with a known value for C, we compute L from the period of oscillation. Damping, and thus R, can be computed from the ratio of successive voltage peaks.

This picture shows normalized curves for damping coefficients of 0.05 and 0.20. Red = capacitor voltage, Blue = discharge current, and Green indicates di/dt (exactly proportional to VL, the voltage across the ideal inductor).
As we turn up the R knob to increase damping: the period increases very slightly, and di/dt progressively leads the capacitor voltage waveform. So the first peak of di/dt waveform is no longer a "level" peak for trivial calculation of damping.

The damping factor in my calibrator is about 0.05, that is to say R = 0.1 * sqrt(L/C).
With a 9V battery for power, and a 555 to trigger a MOSFET, we get bursts of 40 amperes at 15 kHz. Rep rate is about 5 per second, fast enough to get measurements with an analog oscilloscope. Pictures and formulas to follow.

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